## Some Probabilistic Aspects Of Set Partitions (1996)

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Venue: | American Mathematical Monthly |

Citations: | 22 - 2 self |

### BibTeX

@ARTICLE{Pitman96someprobabilistic,

author = {Jim Pitman},

title = {Some Probabilistic Aspects Of Set Partitions},

journal = {American Mathematical Monthly},

year = {1996},

volume = {104},

pages = {201--209}

}

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### Abstract

this paper, section (1.2) offers an elementary combinatorial proof of Dobinski's formula which seems simpler than other proofs in the literature (Rota [35], Berge [5], p. 44, Comtet [9], p. 211). This argument involves identities whose probabilistic interpretations are brought out later in the paper. 1.1 Notation

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Convergence of Probability Measures
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Citation Context ...re identically equal to 1; (ii) the kth moment of X equals B k for every 1sksn. 5 It is well known that for eachs? 0 the Poisson() distribution is uniquely determined by its moments. See for instance =-=[6]-=-, Section 30. So the Poisson(1) distribution is the unique probability distribution whose nth moment is B n for every n. But for each fixed n there are many probability distributions on f0; 1; 2; \Del... |

2044 |
An introduction to probability theory and its applications, volume I
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Citation Context ... \Delta \Delta \Delta n) (19) For X = M n with E[M k n ] j 1 for 0sksn, this simplifies to P (M n = m) = 1 m! n\Gammam X s=0 (\Gamma1) s s! (m = 0; 1; \Delta \Delta \Delta n) (20) See Section IV.4 of =-=[15]-=- for further discussion. According to Proposition (1), the kth moment of M n is B k for every 1sksn. That is to say B k = n X m=1 m k m! n\Gammam X s=0 (\Gamma1) s s! (1sksn) (21) This variation of Do... |

1455 |
An Introduction to Probability Theory
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Citation Context ...: P (X = m) = 1 m! nX k=m (,1) m,k E[X k ] (m , k)! For X = Mn with E[M k n] 1for0 k n, this simpli es to X P (Mn = m) = 1 n,m m! s=0 (,1) s s! (m =0; 1; n) (19) (m =0; 1; n) (20) See Section IV.4 of =-=[15]-=- for further discussion. According to Proposition (1), the kth moment ofMn is Bk for every 1 k n. That is to say Bk = nX m=1 m k m! X n,m s=0 (,1) s s! (1 k n) (21) This variation of Dobinkski's formu... |

508 |
Concrete Mathematics: A Foundation for Computer Science
- Graham, Knuth, et al.
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Citation Context ...ure (Rota [35], Berge [5], p. 44, Comtet [9], p. 211). This argument involves identities whose probabilistic interpretations are brought out later in the paper. 1.1 Notation Following the notation of =-=[20]-=-, let ( n k ) denote the number of partitions of N n into exactly k distinct non-empty subsets, so that B n = n X k=1 ( n k ) (3) The ( n k ) are known as the Stirling numbers of the second kind. Let ... |

322 |
An Introduction to Combinatorial Analysis
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Citation Context ...ences that contain exactly k distinct symbols ([38], p 35.). As an identity of polynomials in m of degree n this identity provides an alternative definition of the coefficients ( n k ) for 1sksn. See =-=[9, 33, 34, 38]-=-. for background and a wealth of further information about Stirling numbers. 1.2 A quick proof of Dobinski's formula. Divide (5) by m! to obtain for positive integer m and n m n m! = n X k=1 ( n k ) 1... |

299 |
Combinatorial Identities
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Citation Context ...ences that contain exactly k distinct symbols ([38], p 35.). As an identity of polynomials in m of degree n this identity provides an alternative definition of the coefficients ( n k ) for 1sksn. See =-=[9, 33, 34, 38]-=-. for background and a wealth of further information about Stirling numbers. 1.2 A quick proof of Dobinski's formula. Divide (5) by m! to obtain for positive integer m and n m n m! = n X k=1 ( n k ) 1... |

176 |
Subadditive processes
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Citation Context ...t in probabilistic notation. This argument differs from the proof in Section 1.2 in that it involves checking (15) fors= 1. Formula (15) has the following interpretation in terms of a Poisson process =-=[24, 31]-=-. Let 0 ! U (1) ! \Delta \Delta \Delta ! U (Xs) ! 1 (16) denote the random locations in (0; 1) of the points of a homogeneous Poisson process on (0; 1) with mean intensity measuresdu for 0 ! u ! 1. Fo... |

148 |
Algebraic Combinatorics
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Citation Context ...ations of Stirling numbers and their generalizations to the computation of moments of probability distributions. Moments of the normal distribution also have interesting combinatorial interpretations =-=[14, 19]-=-. More generally, the idea of representing combinatorially defined numbers by an infinite sum or an integral, typically with a probabilistic interpretation, has proved to be a very fruitful one. Other... |

109 | Asymptotic enumeration methods
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Citation Context ...ch leads to Stirling's formula [7, 12, 27], and Laplace's representation of kth differences of powers [25, 10, 23], which yields an asymptotic formula for the Stirling numbers of the second kind. See =-=[29]-=- for a recent survey of asymptotic enumeration methods. 3 Variations of Dobinski's Formula The derivation of Dobinski's formula given the previous section yields the following proposition: Proposition... |

105 | Cutsem. A calculus for the random generation of labelled combinatorial structures - Flajolet, Zimmermann, et al. - 1994 |

86 |
The number of partitions of a set
- Rota
- 1964
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Citation Context ...ion of nonempty subsets of N n . Let P n denote the set of all such partitions, and let B n = #(P n ), the number of partitions of N n . The numbers B n are known as the Bell numbers [4, 3]. See Rota =-=[35]-=- for a survey of their properties and applications. Dobinski [13] discovered the remarkable formula B n = e \Gamma1 1 X m=1 m n m! (n = 1; 2; \Delta \Delta \Delta) (1) which leads ([26] 1.9) to the as... |

42 |
Principles of Combinatorics
- Berge
- 1971
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Citation Context ...set partitions are the main theme of this paper, section (1.2) offers an elementary combinatorial proof of Dobinski's formula which seems simpler than other proofs in the literature (Rota [35], Berge =-=[5]-=-, p. 44, Comtet [9], p. 211). This argument involves identities whose probabilistic interpretations are brought out later in the paper. 1.1 Notation Following the notation of [20], let ( n k ) denote ... |

31 |
Exponential numbers
- Bell
- 1934
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Citation Context ...unordered collection of nonempty subsets of N n . Let P n denote the set of all such partitions, and let B n = #(P n ), the number of partitions of N n . The numbers B n are known as the Bell numbers =-=[4, 3]-=-. See Rota [35] for a survey of their properties and applications. Dobinski [13] discovered the remarkable formula B n = e \Gamma1 1 X m=1 m n m! (n = 1; 2; \Delta \Delta \Delta) (1) which leads ([26]... |

27 |
Independent process approximations for random combinatorial structures
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Citation Context ...ormulae with binomial coefficients typically involve independent trials, while those with Stirling numbers of the first kind typically involve the cycle structure of random permutations [1]. See also =-=[2]-=- for probabilistic analysis of more general combinatorial structures and further references. 7 4 Random Partitions A random partition of N n is a random variable \Pi with values in the set P n of part... |

18 |
S.: The cycle structure of random permutations
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Citation Context ...rpretations. Formulae with binomial coefficients typically involve independent trials, while those with Stirling numbers of the first kind typically involve the cycle structure of random permutations =-=[1]-=-. See also [2] for probabilistic analysis of more general combinatorial structures and further references. 7 4 Random Partitions A random partition of N n is a random variable \Pi with values in the s... |

14 |
Summierung der Reihe S .... für m
- Dobinski
- 1877
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Citation Context ...ch partitions, and let B n = #(P n ), the number of partitions of N n . The numbers B n are known as the Bell numbers [4, 3]. See Rota [35] for a survey of their properties and applications. Dobinski =-=[13]-=- discovered the remarkable formula B n = e \Gamma1 1 X m=1 m n m! (n = 1; 2; \Delta \Delta \Delta) (1) which leads ([26] 1.9) to the asymptotic evaluation B ns1 p n (n) n+1=2 e (n)\Gamman\Gamma1 as n ... |

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Gaussian and non-gaussian random fields associated with markov processes
- Dynkin
- 1984
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Citation Context ...ations of Stirling numbers and their generalizations to the computation of moments of probability distributions. Moments of the normal distribution also have interesting combinatorial interpretations =-=[14, 19]-=-. More generally, the idea of representing combinatorially defined numbers by an infinite sum or an integral, typically with a probabilistic interpretation, has proved to be a very fruitful one. Other... |

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Exponential polynomials
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Citation Context ...unordered collection of nonempty subsets of N n . Let P n denote the set of all such partitions, and let B n = #(P n ), the number of partitions of N n . The numbers B n are known as the Bell numbers =-=[4, 3]-=-. See Rota [35] for a survey of their properties and applications. Dobinski [13] discovered the remarkable formula B n = e \Gamma1 1 X m=1 m n m! (n = 1; 2; \Delta \Delta \Delta) (1) which leads ([26]... |

10 |
Stirling behavior is asymptotically normal
- Harper
- 1967
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Citation Context ... probabilistically in terms of a uniform random partition, that is a random partition \Pi with the uniform distribution P (\Pi = ) = 1=B n for each partitions2 P n . For developments of this idea see =-=[22, 21, 36, 18]-=-. Random partitions with non-uniform distribution also arise naturally in various contexts. So it is useful to have models for random partitions, both uniform and non-uniform. The following random all... |

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Limit theorems for the number of empty cells in an equiprobable scheme for the distribution of particles by groups. Theory Probab
- Vatutin, Mikhailov
- 1982
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Citation Context ...us contexts. So it is useful to have models for random partitions, both uniform and non-uniform. The following random allocation scheme is the simplest way to generate a random partition of N n . See =-=[10, 40, 41]-=- for extensive study of this and related schemes, and further references. Throw n balls labelled by N n into m boxes labelled by Nm , and assume all m n possible allocations of balls into boxes are eq... |

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Moment Recurrence Relations for Binomial, Poisson and Hypergeometric Frequency Distributions
- Riordan
- 1937
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Citation Context ...e that the right side of (8) equals E[X ns]. So the identity (8) amounts to the formula E[X ns] = n X k=1 ( n k ) k (n = 1; 2; \Delta \Delta \Delta) (14) for the moments of the Poisson() distribution =-=[32, 30]-=-. This formula is the particular case of (11) for X with Poisson() distribution, for it is known [32, 10] that E[X ks] =sk (k = 1; 2; \Delta \Delta \Delta ) (15) Formula (15) follows easily from (13) ... |

4 |
Random partitions of sets. Theory Probab
- Sachkov
- 1973
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Citation Context ... probabilistically in terms of a uniform random partition, that is a random partition \Pi with the uniform distribution P (\Pi = ) = 1=B n for each partitions2 P n . For developments of this idea see =-=[22, 21, 36, 18]-=-. Random partitions with non-uniform distribution also arise naturally in various contexts. So it is useful to have models for random partitions, both uniform and non-uniform. The following random all... |

3 |
A note on easy proofs of Stirling's theorem
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- 1986
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Citation Context ...or an integral, typically with a probabilistic interpretation, has proved to be a very fruitful one. Other examples are the representation of n! as a gamma integral, which leads to Stirling's formula =-=[7, 12, 27]-=-, and Laplace's representation of kth differences of powers [25, 10, 23], which yields an asymptotic formula for the Stirling numbers of the second kind. See [29] for a recent survey of asymptotic enu... |

3 | An elementary proof of Stirling's formula
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- 1986
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Citation Context ...or an integral, typically with a probabilistic interpretation, has proved to be a very fruitful one. Other examples are the representation of n! as a gamma integral, which leads to Stirling's formula =-=[7, 12, 27]-=-, and Laplace's representation of kth differences of powers [25, 10, 23], which yields an asymptotic formula for the Stirling numbers of the second kind. See [29] for a recent survey of asymptotic enu... |

3 |
The structure of partitions of large sets
- Fristedt
- 1987
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Citation Context ... probabilistically in terms of a uniform random partition, that is a random partition \Pi with the uniform distribution P (\Pi = ) = 1=B n for each partitions2 P n . For developments of this idea see =-=[22, 21, 36, 18]-=-. Random partitions with non-uniform distribution also arise naturally in various contexts. So it is useful to have models for random partitions, both uniform and non-uniform. The following random all... |

3 |
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- Haigh
- 1972
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On numbers related to partitions of unlike objects and occupancy problems
- Holst
- 1981
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Citation Context ...ed to be a very fruitful one. Other examples are the representation of n! as a gamma integral, which leads to Stirling's formula [7, 12, 27], and Laplace's representation of kth differences of powers =-=[25, 10, 23]-=-, which yields an asymptotic formula for the Stirling numbers of the second kind. See [29] for a recent survey of asymptotic enumeration methods. 3 Variations of Dobinski's Formula The derivation of D... |

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Theorie Analytique des Probabilites (2nd Ed
- Laplace
- 1814
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Citation Context ...ed to be a very fruitful one. Other examples are the representation of n! as a gamma integral, which leads to Stirling's formula [7, 12, 27], and Laplace's representation of kth differences of powers =-=[25, 10, 23]-=-, which yields an asymptotic formula for the Stirling numbers of the second kind. See [29] for a recent survey of asymptotic enumeration methods. 3 Variations of Dobinski's Formula The derivation of D... |

3 |
A method and two algorithms in the theory of partitions
- Nijenhuis, Wilf
- 1975
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Citation Context ...ibution of \Pi nM defined by formula (36) defines a Gibbs' measure on partitions of N n . See Steele [39] for further discussion of such measures on combinatorial objects. See also Nijenhuis and Wilf =-=[28]-=- for a 10 recursive algorithm for construction of a uniform random partition of N n based on the recurrence B n = 1 + n\Gamma1 X k=1 / n \Gamma 1 k ! B k (38) where the right side counts the number of... |

3 |
Generation of a random partition of a set by an urn model
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- 1983
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Citation Context ... (29) Because the probability displayed in (28) depends on the number of occupied boxes k, this random partition \Pi of N n has a non-uniform distribution for all n; ms2. However, as observed by Stam =-=[37]-=-, for each fixed n it is possible to generate a uniformly distributed random partition \Pi of N n by a suitable 8 randomization of m. The following Proposition was suggested by Stam's construction, wh... |

3 |
Gibbs' measures on combinatorial objects and the central limit theorem for an exponential family of random trees
- Steele
- 1987
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Citation Context ...ksn) (37) so the fact that these probabilities sum to 1 amounts to formula (14) forsn (). The distribution of \Pi nM defined by formula (36) defines a Gibbs' measure on partitions of N n . See Steele =-=[39]-=- for further discussion of such measures on combinatorial objects. See also Nijenhuis and Wilf [28] for a 10 recursive algorithm for construction of a uniform random partition of N n based on the recu... |

3 |
Gaussian and non-Gaussian random elds associated with Markov processes
- Dynkin
- 1984
(Show Context)
Citation Context ...ations of Stirling numbers and their generalizations to the computation of moments of probability distributions. Moments of the normal distribution also have interesting combinatorial interpretations =-=[14, 19]-=-. More generally, the idea of representing combinatorially de ned numbers by an in nite sum or an integral, typically with a probabilistic interpretation, has proved to be a very fruitful one. Other e... |

2 |
A simple derivation of Stirling's asymptotic series
- Namias
- 1986
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Citation Context ...or an integral, typically with a probabilistic interpretation, has proved to be a very fruitful one. Other examples are the representation of n! as a gamma integral, which leads to Stirling's formula =-=[7, 12, 27]-=-, and Laplace's representation of kth differences of powers [25, 10, 23], which yields an asymptotic formula for the Stirling numbers of the second kind. See [29] for a recent survey of asymptotic enu... |

1 |
Charalambides and Jagbir Singh. A review of the Stirling numbers, their generalizations and statistical applications
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- 1988
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Citation Context ...lta \Delta \Delta ( du k ) du 1 \Delta \Delta \Delta du k =sk (17) So the expected number of points in the k-tuple point process is E[X ks] =sk Z 1 0 du 1 \Delta \Delta \Delta Z 1 0 du k =sk (18) See =-=[33, 8]-=- for various applications of Stirling numbers and their generalizations to the computation of moments of probability distributions. Moments of the normal distribution also have interesting combinatori... |

1 |
Counting subsets of the random partition and the `Brownian bridge' process
- DeLaurentis, Pittel
- 1983
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Citation Context ...e n of a uniform random partition of N n to be deduced from corresponding results for the classical occupancy problem defined by random allocations of balls in boxes, for which see [40, 41]. See also =-=[22, 21, 36, 11, 18, 2]-=- for a more detailed account of the asymptotics of uniform random partitions of N n . As a variation, the following Corollary is easily obtained by a similar argument: Corollary 5 Suppose that M has t... |

1 |
A note on the moments of a Poisson probability distribution
- Philipson
- 1963
(Show Context)
Citation Context ...e that the right side of (8) equals E[X ns]. So the identity (8) amounts to the formula E[X ns] = n X k=1 ( n k ) k (n = 1; 2; \Delta \Delta \Delta) (14) for the moments of the Poisson() distribution =-=[32, 30]-=-. This formula is the particular case of (11) for X with Poisson() distribution, for it is known [32, 10] that E[X ks] =sk (k = 1; 2; \Delta \Delta \Delta ) (15) Formula (15) follows easily from (13) ... |

1 |
Random partitions of sets
- Sachkov
- 1973
(Show Context)
Citation Context ...n be phrased probabilistically in terms of a uniform random partition, that is a random partition with the uniform distribution P ( = )=1=Bn for each partition 2 Pn. For developments of this idea see =-=[22, 21, 36, 18]-=-. Random partitions with non-uniform distribution also arise naturally in various contexts. So it is useful to have models for random partitions, both uniform and non-uniform. The following random all... |